Average Error: 13.5 → 0.3
Time: 8.9s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot {\left({\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{2}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{\cos \left(e^{\log \left(2 \cdot \pi\right) + \log u2}\right)} \]
\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)
↓
\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot {\left({\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{2}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{\cos \left(e^{\log \left(2 \cdot \pi\right) + \log u2}\right)}

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (*
   (sqrt (- (log1p (- u1))))
   (pow (pow (cos (* 2.0 (* PI u2))) 2.0) 0.3333333333333333))
  (cbrt (cos (exp (+ (log (* 2.0 PI)) (log u2)))))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf(1.0f - u1)) * cosf((2.0f * ((float) M_PI)) * u2);
}
↓
float code(float cosTheta_i, float u1, float u2) {
	return (sqrtf(-log1pf(-u1)) * powf(powf(cosf(2.0f * (((float) M_PI) * u2)), 2.0f), 0.3333333333333333f)) * cbrtf(cosf(expf(logf(2.0f * ((float) M_PI)) + logf(u2))));
}

Error

Bits error versus cosTheta_i

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.5

    \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

  2. Simplified0.3

    \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]

  3. Applied add-exp-log_binary320.3

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot \color{blue}{e^{\log u2}}\right) \]

  4. Applied add-exp-log_binary320.4

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\color{blue}{e^{\log \left(2 \cdot \pi\right)}} \cdot e^{\log u2}\right) \]

  5. Applied prod-exp_binary320.3

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(e^{\log \left(2 \cdot \pi\right) + \log u2}\right)} \]

  6. Applied add-cube-cbrt_binary320.5

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(e^{\log \left(2 \cdot \pi\right) + \log u2}\right)} \cdot \sqrt[3]{\cos \left(e^{\log \left(2 \cdot \pi\right) + \log u2}\right)}\right) \cdot \sqrt[3]{\cos \left(e^{\log \left(2 \cdot \pi\right) + \log u2}\right)}\right)} \]

  7. Applied associate-*r*_binary320.5

    \[\leadsto \color{blue}{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\sqrt[3]{\cos \left(e^{\log \left(2 \cdot \pi\right) + \log u2}\right)} \cdot \sqrt[3]{\cos \left(e^{\log \left(2 \cdot \pi\right) + \log u2}\right)}\right)\right) \cdot \sqrt[3]{\cos \left(e^{\log \left(2 \cdot \pi\right) + \log u2}\right)}} \]

  8. Simplified0.4

    \[\leadsto \color{blue}{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \cdot \sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)\right)} \cdot \sqrt[3]{\cos \left(e^{\log \left(2 \cdot \pi\right) + \log u2}\right)} \]

  9. Taylor expanded in u2 around inf 0.3

    \[\leadsto \left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{{\left({\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{2}\right)}^{0.3333333333333333}}\right) \cdot \sqrt[3]{\cos \left(e^{\log \left(2 \cdot \pi\right) + \log u2}\right)} \]

  10. Final simplification0.3

    \[\leadsto \left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot {\left({\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{2}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{\cos \left(e^{\log \left(2 \cdot \pi\right) + \log u2}\right)} \]

Reproduce

herbie shell --seed 2021329 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))